The minimal evolution time between two distinguishable states is of fundamental interest in quantum physics. Very recently Mirkin et al. [Phys. Rev. A 94, 052125 (2016)] argued that some of the most common quantum-speed-limit (QSL) bounds which depend on the actual evolution time do not cleave to the essence of the QSL theory as they grow indefinitely but the final state is reached at a finite time in a damped Jaynes-Cummings model. In this paper, we thoroughly study this puzzling phenomenon. We find the inconsistent estimates will happen if and only if the limit of resolution of a calculation program is achieved, through which we propose that the nature of the inconsistency is not a violation of the essence of the QSL theory but an illusion caused by the finite precision in numerical simulations. We also present a generic method to overcome the inconsistent estimates and confirm its effectiveness in both amplitude-damping and phase-damping channels. Additionally, we show special cases which may restrict the QSL bound defined by ``quantumness''.
The minimal evolution time between two distinguishable states is of fundamental interest in quantum physics. Very recently Mirkin et al. [Phys. Rev. A 94, 052125 (2016)] argued that some of the most common quantum-speed-limit (QSL) bounds which depend on the actual evolution time do not cleave to the essence of the QSL theory as they grow indefinitely but the final state is reached at a finite time in a damped Jaynes-Cummings model. In this paper, we thoroughly study this puzzling phenomenon. We find the inconsistent estimates will happen if and only if the limit of resolution of a calculation program is achieved, through which we propose that the nature of the inconsistency is not a violation of the essence of the QSL theory but an illusion caused by the finite precision in numerical simulations. We also present a generic method to overcome the inconsistent estimates and confirm its effectiveness in both amplitude-damping and phase-damping channels. Additionally, we show special cases which may restrict the QSL bound defined by ``quantumness''.
Quantum mechanics dictates bounds for the minimal evolution time between predetermined initial and final states. Several of these quantum-speed-limit (QSL) bounds were derived for nonunitary dynamics using different approaches. Here, we perform a systematic analysis of the most common QSL bounds in the damped Jaynes-Cummings model, covering the Markovian and non-Markovian regimes. We show that only one of the analyzed bounds cleaves to the essence of the QSL theory outlined in the pioneer works of Mandelstam and Tamm and of Margolus and Levitin in the context of unitary evolutions. We also show that all QSL bounds analyzed reflect the fact that in our model non-Markovian effects speed up quantum evolution. However, it is not possible to infer Markovian or non-Markovian behavior of the dynamics by analyzing only the QSL bounds.
The authors of the comment[Phys. Rev. A 97, 046101 (2018)] raise that the inconsistency in calculating some common quantum-speed-limit (QSL) bounds, which is presented in our paper [Phys. Rev. A 95, 052118 (2017)], does not exist in their paper [Phys. Rev. A 94, 052125 (2016)]. Therefore they insist that their criticism to some QSL bounds is still valid. We demonstrate all the QSL bounds mentioned in the comment are similar in essence. We also show the inconsistency which is presented in their original paper can not be used to support their criticism. Furthermore, we exhibit although the inconsistency presented in our paper is not a same one to which is presented by the authors of the comment, it exists and be unavoidable in their example. Accordingly, we believe the result in our paper is usable in numerical calculations.
In a recent paper [Phys. Rev. A 95, 052118 (2017)], the authors claim that our criticism, in Phys. Rev. A 94, 052125 (2016), to some quantum speed limit bounds for open quantum dynamics that appeared recently in literature are invalid. According to the authors, the problem with our analysis would be generated by an artifact of the finite-precision numerical calculations. We analytically show here that it is not possible to have any inconsistency associated with the numerical precision of calculations. Therefore, our criticism of the quantum speed limit bounds continues to be valid.
The minimal time a system needs to evolve from an initial state to its one orthogonal state is defined as the quantum speed limit time, which can be used to characterize the maximal speed of evolution of a quantum system. This is a fundamental question of quantum physics. We investigate the generic bound on the minimal evolution time of the open dynamical quantum system. This quantum speed limit time is applicable to both mixed and pure initial states. We then apply this result to the damped Jaynes-Cummings model and the Ohimc-like dephasing model starting from a general time-evolution state. The bound of this time-dependent state at any point in time can be found. For the damped Jaynes-Cummings model, when the system starts from the excited state, the corresponding bound first decreases and then increases in the Markovian dynamics. While in the non-Markovian regime, the speed limit time shows an interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing model, this bound would be gradually trapped to a fixed value. In addition, the roles of the relativistic effects on the speed limit time for the observer in non-inertial frames are discussed.
As one of the fundamental questions, quantum speed limit (QSL) is of vital role in the community of quantum physics, which sets the ultimate, maximal speed with which a quantum system can evolve. QSL originated from the understanding of the Heisenberg uncertainty relation. In 1945, Mandelstam and Tamm found the time Δt in the celebrated relation ΔEΔt≥ℏ could be thought as the minimum time that a quantum system needs to evolve from an initial state to the orthogonal state for unitary evolution. QSL sets the lower bound of the evolution. They established the relationship between the minimum evolution time ( τQSL ) and the energy uncertainty of the system (ΔE), namely τQSL=πℏ/(2ΔE) , the Mandelstam-Tamm (MT) bound. Later, Margolus and Levitin derived another different bound for a closed system which connected the τQSL with the average energy of the system ( 〈E〉 ), i.e., τQSL=πℏ/(2〈E〉) , the Margolus-Levitin (ML) bound. Combining the MT and ML bounds together, we can have a unified bound for a unitary system and orthogonal states. In the case of open system dynamics, because there is energy and/or coherence exchange between the system and the environment, the orthogonal final state of the initial state can hardly be reached any longer. An investigation of the QSL on non-unitary quantum evolution needs to be considered. The generalized QSLs are established by employing the distance between the distinguishable states. Different metrics define different distances, leading to different QSL bounds. We show a few typical generalized QSL bounds based on different metrics. And the unified QSL bound for non-unitary evolution is obtained by combining the generalized MT and ML bounds under the same metric. The generations and developments of QSL have successfully generalized the MT bounds. Even the ML bounds have been attempted to generalize, the current studies of ML bound are thought as the significantly harder task, and keep as the open questions. The influences of external environment and quantum system dynamics which include the role of entanglement in QSL for open dynamics and many-body systems, the relation between the maximum interaction speed and QSL in quantum spin systems, the non-Markovianity effect of the environment on accelerating the speed of evolution have been investigated. Furthermore, by employing the gauge invariant distance in the Riemannian metric, we establish a distinct QSL bound, which relates the quantum speed limit time to the geometry phase, and presents the geometric and gauge invariant properties of it. The bound shows the impacts of the transmission modes of the state vectors on the evolution path in the manifold. Different from the established MT and ML bounds, our bound establishes the relation between the QSL and the Pancharatnam connection 1-form, and our results show that the QSL can be controlled via manipulating the generalized Pahcharatnam connection. Also, our results of QSL are holonomy via the generalized Pahcharatnam connection, which show us a possible technique to speed up the holonomic quantum computation. The bound is also interesting to the metrology and non-equilibrium thermodynamics, etc. QSL has also been studied in the quantum representation of Wigner function. In the future research of quantum speed limit, there are many issues that need to be considered, such as the generalized ML bound, the unified QSL bound under different metrics, the application of maximum rate of quantum information derived from the QSL, and the relation between the quantum speed limit and the quantum heat engine, etc.
We develop an intuitive geometric picture of quantum states, define a particular state distance, and derive a quantum speed limit (QSL) for open systems. Our QSL is attainable because any initial state can be driven to a final state by the particular dynamics along the geodesic. We present the general condition for dynamics along the geodesic for our QSL. As evidence, we consider the generalized amplitude damping dynamics and the dephasing dynamics to demonstrate the attainability. In addition, we also compare our QSL with others by strict analytic processes as well as numerical illustrations, and show our QSL is tight in many cases. It indicates that our paper is significant in tightening the bound of evolution time.